Collaborative Statistics Teacher's Guide

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Collaborative Statistics Teachers Guide

By:

Barbara Illowsky, Ph.D.

Susan Dean


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Collaborative Statistics Teachers Guide

By:

Barbara Illowsky, Ph.D.

Susan Dean

Online:< http://cnx.org/content/col10547/1.5/ >

C O N N E X I O N S

Rice University, Houston, Texas


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This selection and arrangement of content as a collection is copyrighted by Maxfield Foundation. It is licensed under

the Creative Commons Attribution 2.0 license (http://creativecommons.org/licenses/by/2.0/).

Collection structure revised: October 1, 2008

PDF generated: February 4, 2011

For copyright and attribution information for the modules contained in this collection, see p. 50.


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1 Suggested Plan for Teaching the Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Ch. 1: Sampling and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Ch. 2: Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Ch. 3: Probability Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Ch. 4: Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Ch. 5: Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 CH. 6: Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Ch. 7: Central Limit Theroem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Ch. 8: Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710 Ch. 9: Hypothesis Testing of Single Mean and Single Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111 Ch 10: Hypothesis Testing of Two Means and Two Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312 Ch 11: The Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513 Ch 12: Linear Regression and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114 Ch 13: F Distribution and ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 9Attributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0


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Chapter 1

Suggested Plan for Teaching the Course1

Each chapter isinteractive. Students should fill in the blanks and answer the questions.

At the end of each chapter is at least one practice. The practice leads the students step-by-step throughproblems. We, the authors, start the practices in calss with students working in groups of 2, 3, or 4. Thestudents finish the practices at home. The practiceis after the chapter reading but before the homework.

The back of the book contains answers to the odd-numbered homework problems. In this plan (thisdocument), the suggested homework is listed at the end of the chapter discussion.

At the end of each chapter (after the homework), there is at least one lab. The labs use real data collectedby the instructor or the students or both. We often use the class to collect data. Labs may be done in groupsand are an excellent teaching tool especially if they are started in class. The book contains the followinglabs:

Ch. 1: Data Collection Lab I (number of movies viewed)

Ch. 1: Sampling Experiment Lab II (table of restaurants provided)Ch. 2: Descriptive Statistics Lab (number of pairs of shoes)Ch. 3: Probability Lab (counting M&Ms)Ch. 4: Discrete Distribution Lab I (picking playing cards)Ch. 4: Discrete Distribution Lab II (Tet game)Ch. 5: Continuous Distribution Lab (generate random numbers)Ch. 6: Normal Distribution Lab I (Terry Vogels lap times provided)Ch. 6: Normal Distribution Lab II (measure pinkie fingers)Ch. 7: Central Limit Theorem Lab I (counting change)Ch. 7: Central Limit Theorem Lab II (cookie recipes)Ch. 8: Confidence Interval Lab I (real estate prices)Ch. 8: Confidence Interval Lab II (students born in state)Ch. 8: Confidence Interval Lab III (heights of women)

Ch. 9: Hypothesis Testing Lab - Single Mean and Single Proportion (3 tests)Ch. 10: Hypothesis Testing Lab - Two Means and Two Proportions (3 tests)Ch. 11: Chi-Square Goodness of Fit Lab I (grocery store receipts)Ch. 11: Chi-Square Test for Independence Lab II (favorite snack/gender)Ch. 12: Regression Lab I (distance from school vs. cost of supplies this term)Ch. 12: Regression Lab II (number of pages in textbook vs. cost of textbook)Ch. 12: Regression Lab II (weights vs. fuel efficiency)Ch. 13: ANOVA Lab (fruits, vegetables, breads)

1This content is available online at .

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2 CHAPTER 1. SUGGESTED PLAN FOR TEACHING THE COURSE

Because the authors use technology heavily in the course (making many class periods a lab), we typicallychoose to do 6 labs during the quarter. The labs are best done in groups of 2, 3, or 4.

There are five projects in the book. The Univariate Data project covers the ideas in chapters 1 and 2.

The Continuous Distributions and Central Limit Theorem project covers idea in chatters 5, 6, and 7. TheHypothesis Testing - Article and the Hypothesis Testing - Word project covers ideas in chapters 8 and 9.The Bivariate Data, Linear Regression and Univariate project covers ideas in chapters 1, 2, and 12. Projectsare done in groups of 2, 3, or 4.

There arePractice Finalswith answers andData Setsin the text. One of the Chapter 6 Labs uses one of thedata sets. Going over the Table of Contents for this collection with the students is recommended.

We carry probabilities to 4 decimal places.

The number of days (a "day" is a 50 minute period) based on a quarter system (10 weeks of class, 1 weekof finals) it takes to cover a chapter is below. At De Anza, we are on a quarter system. In a semester, youcould spend more time analyzing real data. The material is meant to be covered in one quarter or in one

semester. Introduction - 2 days Descriptive Statistics - 4 days Probability Topics - 4 days Discrete Random Variables - 5 days Continuous Random Variables - 3 days The Normal Distribution - 3 days The Central Limit Theorem - 3 days Confidence Intervals - 4 days Hypothesis Testing - Single Mean and Single Proportion - 4 days Hypothesis Testing - Two Means and Two Proportions - 4 days The Chi-Square Distribution - 4 days

Linear Regression and Correlation - 4 days

Analysis of Variance and F Distribution - 3 days


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Chapter 2

Ch. 1: Sampling and Data1

Explain the terms statistics and probability.

Introduce the key terms by an example.

Example 2.1Students may be interested in the average time (in years) it will take them to earn a B.A. or B.S.

Differentiate between population and sample.

Explain data. The book discusses qualitative and quantitative data. Quantitative data is either discrete(countable) or continuous (measurable).

Types of Data

Qualitative data- the city or town a student lives in. Quantitative discrete (countable) data- the number of T-shirts a student owns. Quantitative continuous (measurable) data - the amount of time (in hours) a student studies statistics

each day.

SamplingDiscuss what a sample is. Stress the importance of sampling randomly and the fact that two randomsamples from the same population may be different. Doing the two experiments with a fair die (roll thesame die 20 times for each experiment and record the frequencies of the faces in the book) will help themunderstand how samples vary. Using your class as the population, sample 10 men and 10 women. Let thesample be the number of pairs of shoes each student owns. This example illustrates samples which arenotrepresentative from the same population.

Discuss how to sample data. Though there are numerous ways, the book discusses simple random, strati-fied, cluster, systematic, and convenience. You may want to discuss other ways of sampling.

FrequencyThe last part of the chapter discusses frequency, relative frequency, and cumulative relative frequency. Thestudents should understand how to read the table in the example (heights, to the nearest inch, of malestudents at ABC College).

Assign PracticeTake some class time and have the students work in groups and complete the Practice2.

1This content is available online at .2"Sampling and Data: Practice 1"

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4 CHAPTER 2. CH. 1: SAMPLING AND DATA

Assign HomeworkAssign Homework3 problems: 1 - 17 odds, 19 - 27.

3"Sampling and Data: Homework"


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Chapter 3

Ch. 2: Descriptive Statistics1

Graphs are important tools in statistics and probability. Graphs used in this course are the boxplot, thehistogram, and the stem-plot. The histogram and boxplot are used extensively while the stem-plot is justdemonstrated.

Illustrate Examples

To illustrate stem-plots, have the students complete Example 2-2 by hand. To illustrate histograms, have the students do Example 2-4 by hand and then, if you are using tech-

nology, have them do the same example. They can verify their results by looking at the picture. Right after Example 2-4, there is an "Optional Collaborative Classroom Exercise" for the students to

do that involves the amount of money they have in their pocket or purse. To illustrate the boxplot, have the students do Example 2-6. In this example, they will compare two

boxplots.

Center of DataDiscuss the measures of "center" - mean (average), median, mode. If you are using technology, it helpsto show the students how to use technology to find the measures first. Then do some examples by hand.Distinguish between the symbols used for the sample mean and the population mean. Give an examplewhere the mean is the best measure of the center and a second example where the median is the bestexample. (Example where median is the better measure: 19, 16, 46, 18, 21. Example where mean is the

better measure: 18, 20, 23, 25, 25.) At the end of the chapter, there is a summary of the mean formulas if youdesire to go over them.

Spread of DataDiscuss the measures of spread - variance and standard deviation. Stress that the standard deviation isthe square root of the variance. Differentiate between the sample and population standard deviations.Dividing byn 1 in the sample variance formula makes the sample standard deviation a better estimatorof the population standard deviation. Do one example by hand and have the students participate (the set{1, 2, 3} is quick and easy). They will have to calculate the mean first. They should discover how easy it is

to make a numerical error when they calculate standard deviation by hand.

Location of DataDiscuss the measures of location - quartile and percentile. For many students, these measures are difficult.It is better to make up a relative frequency table from an example like the one in the book (the amount ofsleep 50 students get per school night) and find quartiles and percentiles. Graphing calculators typicallycalculate quartiles.


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6 CHAPTER 3. CH. 2: DESCRIPTIVE STATISTICS

Definition of ValueWe introduce the formula Value= Mean + (#ofSTDEVs)(Standard Deviation) in this chapter. For example,a student with a 74 on the first exam in a statistics class wants to compare his score to a student who receiveda 70 in another section. If the mean and standard deviation for the first class was 72 and 4, respectively, andthe mean and standard deviation for the second class was 68 and 2, respectively, which student did betterrelative to the class? Solve the equation for #OfSTDEVs in each case.

Assign PracticeHave students work in groups to complete Practice 12 and Practice 23.

Calculator InstructionsIf you are using the TI-83 or TI-84 calculator series, go over the calculator instructions in the text for en-tering data and calculating the sample mean, the sample standard deviation, the quartiles, constructinghistograms, and construction boxplots. The calculator instructions can also be found on the Texas Instru-ments website and the appropriate Guidebook.

Assign Homework

Assign Homework4

. Suggested problems: 1 - 23 odds, 24 - 30.

2"Descriptive Statistics: Practice 1" 3"Descriptive Statistics: Practice 2" 4"Descriptive Statistics: Homework"


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Chapter 4

Ch. 3: Probability Topics1

The best way to introduce the terms is through examples. You can introduce the terms experiment, out-come, sample space, event, probability, equally likely, conditional, mutually exclusive events, and indepen-dent events AND you can introduce the addition rule, the multiplication rule with the following example:In a box (you cannot see into it), there are are 4 red cards numbered 1, 2, 3, 4 and 9 green cards numbered 1,2, 3, 4, 5, 6, 7, 8, 9. You randomly draw one card (experiment). LetR be the event the card is red. LetG bethe event the card is green. Let Ebe the event the card has an even number on it.

Example 4.1

Event Card Example

1. List all possible outcomes (the sample space). Have students list the sample space in theform {R1, R2, R3, R4, G1, G2, G3, G4, G5, G6, G7, G8, G9}. Each outcome is equally likely.

Plane outcome = 113 .2. FindP (R).3. FindP (G). G is the complement of R.P (G)+P (R)= _______.

4. P ( red card given a that the card has an even number on it) = P (R | E).This is a conditional.Pick the red card out of the even cards. There are 6 even cards.

5. FindP (R ANDE). (Multiplication Rule: P (R and E)= P (E | R) (P (R) )6. P (R OR E). (Addition Rule: P (R OR E)= P (E)+P (R) P (E ANDR))7. Are the eventsRandGmutually exclusive? Why or why not?8. Are the eventsGand Eindependent? Why or why not?

Example 4.2(Optional Topic)AVenn diagramis a tool that helps to simplify probability problems. Introduce

a Venn diagram using an example. Example: Suppose 40% of the students at ABC College belongto a club and 50% of the student body work part time. Five percent of the student body works parttime and belongs to a club.

Have the students work in groups to draw an appropriate Venn diagram after you have shownthem what a Venn diagram basically looks like. The diagram should consist of a rectangle withtwo overlapping circles. One rectangle represents the students who belong to a club (40%) and theother circle represents those students who work part time (50%). The overlapping part are thosestudents who belong to a club and who work part time (5%).

Find the following:

1. P(student works part time but does not belong to a club)


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8 CHAPTER 4. CH. 3: PROBABILITY TOPICS

2. P(student belongs to a club given that the student works part time)3. P(student does not belong to a club)4. P(works part time given that the student belongs to a club)5. P(student belongs to a club or the student works part time)

Solution

Figure 4.1

C - student belongs to a clubPT - student works part time

Example 4.3

Find the following:1. P(a child is 9 - 11 years old)2. P(a child prefers regular soccer camp)3. P(a child is 9 - 11 years old and prefers regular soccer camp)4. P(a child is 9 - 11 years old or prefers regular soccer camp)5. P(a child is over 14 given that the child prefers micro soccer camp)6. P(a child prefers micro soccer camp given that the child is over 14)

Tree Diagrams (Optional Topic)A tree is another probability tool. Many probability problems are simplified by a tree diagram. To exemplifythis, suppose you want to draw two cards, one at a time, without replacementfrom the box of 4 red cards

and 9 green cards.


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Figure 4.2: There are (13)(12) = 156 Possible Outcomes. (ex. R1R1, R1R2, R1G3, G3G4, etc.)

Example 4.4

Find the following:

1. P(RR)2. P(RG or GR)3. P(at most one G in two draws)4. P(G on the 2nd draw|R on the 1st draw). The size of the sample space has been reduced to

12 + 36= 481.5. P(no R on the 1st draw)

Introducecontingency tablesas another tool to calculate probabilities. Lets suppose an owner of a soccer

camp for children keeps information concerning the type of soccer camp the children prefer and their ages.The data is for 572 children.

Type of Soccer Camp Preference Under 6 6-8 9-11 12-14 Over 14 Row Total

Micro 42 76 46 25 10 199

Regular 8 68 92 105 100 373

Column Total 50 144 138 130 110 572

Table 4.1

Assign PracticeAssign Practice 12 and Practice 23 in class. Have students work in groups.

Assign LabThe Probability Lab is an excellent way to cement many of the ideas of probability. The lab is a group effort(3 - 4 students per group).

Assign HomeworkAssign Homework4. Suggested problems: 1 - 15 odds, 19, 20, 21, 23, 27, 28 - 30.

2"Probability Topics: Practice" 3"Probability Topics: Practice II" 4"Probability Topics: Homework"


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10 CHAPTER 4. CH. 3: PROBABILITY TOPICS


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Chapter 5

Ch. 4: Discrete Random Variables1

This chapter introduces expected value (long term average) and four of the common discrete randomvariables(binomial, geometric, hypergeometric, and Poisson). The authors cover expected value and two

of the discrete random variables (binomial and Poisson). Depending on your background, you may wantto cover the binomial (usually required) together with none or some of the other discrete random variables

Random VariablesExplain random variable (assigns numerical values to the outcomes of a statistical experiment). Upper caseletters denote random variables. Example: LetX= the number of cars in your household. (The phrase "thenumber of" tells you that Xtakes on discrete values.) Xtakes on the values 0, 1, 2, 3, ...

The Probability Distribution FunctionAprobability distribution function (pdf)is best shown with an example: A controversial drug is given totwo patients. Let X = the number of patients cured.

P(a cure)= 56 P(no cure)=

16

A pdf is easiest to understand in a table.

X P (X)orP (X= x)

0 P (X= 0)=

16

16

=

136

1 P (X= 1)= 2

16

56

=

1036

2 P (X= 2)=

56

56

=

2536

Table 5.1: Each probability is between 0 and 1.

The previous example can be used as an example ofexpected valueor long term average ( ). Make a thirdcolumn labeled ( x) (P (x)). Calculate the three values and add them. The result,(0)

136

+(1)

1036

+

(2)

2536

= 6036 = 1.67, is the expected number of patients who are cured if the drug is administered many

times to two patients.

The binomial is a special discrete pdf or pattern. A binomial experiment consists of counting the numberof successes in one or more Bernoulli trials. (A Bernoulli trial has only two possible outcomes, success orfailure. In every Bernoulli trial, the probability of a success (or failure) remains the same.)


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12 CHAPTER 5. CH. 4: DISCRETE RANDOM VARIABLES

Example 5.1John comes to his stat class and discovers he must take a true-false quiz . There are 20 questionson the quiz. John has not attended class recently and must guess randomly at the questions. LetX= the number of questions John answers correctly out of 20 questions. Xtakes on the values0, 1, 2, 3, ..., 20. P (correct answer: a success)= 0.5. Johns guessing at the answers is a binomialexperiment.

Notation: XB (20, 0.5)where the number of trials, n , is 20 and the probability of a success, p,on any trial is 0.5.

Students can find the mean ( = n p), and the standard deviation (=square root ofn pq) eitherby hand or with technology. (q is the probability of a failure.) Have students help you fill in theblanks and answer the questions:

1. =2. Draw the graph. (horizontal axis is the number of successes; vertical is the probability of 0

successes, 1 success, 2 successes, ..., 20 successes. Draw vertical lines or boxes.

3. What is the probability that John gets 15 questions correct?P (X= 15) More than 15 questions correct? P (X> 15) At least 15 questions correct?(P (X= 15)+P (X> 15))

A geometric experiment takes place when at least one Bernoulli trial is performed and all are failuresexcept the last one which is the only success. Example: Liz likes to play darts. The probability that she hitsthe bulls eye (success) on any throw is 85%. (Liz is good!) Liz throws darts at the bulls eyeuntilshe hitsit. LetX= the number of times Liz throws the dart at the bulls eye until she hits it. Have students help youfill in the blanks:

Fill in the blanks.

X

_______ (X

G (p)wherep = probability of a success= 0.85)

Draw the graph. (Number of throws until the first success versus probability) 4. What is the probability that Liz hits the bulls eye for the first time on the third throw? That ittakes more than three throws for Liz to hit the bulls eye for the first time? That it takes at least threethrows?

Xtakes on the values _______. = _______. In words,is _______

The Geometric EquationP (X= x)= qx1 pHypergeometric DistributionThe hypergeometric distribution is characterized by choosing a sample without replacement from twodistinct groups. One of the two groups is what is of interest in the sample. Some lotteries are based on the

hypergeometric distribution. click to edit noteExample 5.2Suppose a shipment of 20 tape recorders contains 5 defectives. An inspector randomly chooses 8of the tape recorders to inspect. He is interested in the number of defectives in the sample of 8.Have the class answer questions similar to those for the binomial and the geometric.

NOTATION: X H(r, b, n)wherer= size of the group of interest, b= size of the other group, andn= size of the sample.


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Poisson DistributionThe Poisson distribution is concerned with the number of times an event takes place in a certain interval. Itis used in the field of reliability. The Poisson approximates the binomial when n is "large" (say, more than100) and p is "small" (say, less than 0.1).

Example 5.3Suppose the average number of accidents that occur in a week at a particularly busy intersection

is one. The interval is one week. The average is one accident. LetX= the number of accidentsthat occur in a one week period at the intersection. Have the students help fill in the blanks andanswer the questions:

1. X _______ (X P ( where = one accident)2. What values doesXtake on?3. What is the probability that at most one accident occurs in a week?

The Poisson Distribution Formula

The parameter for the Poisson is the mean,. Some books and calculators use the Greek letter, (lambda)as the mean. The equation for the Poisson is:

P (X= x)=x e

x! where x= 0,1,2,3,... (5.1)

Assign PracticeHave the students complete theportion of the practicethat is appropriate for what you have covered inclass. Expected Value, Binomial, and Poisson are dealt with Practice 12, Practice 23, and Practice 34. Practice45 is based on the Geometric Distribution, while Practice 56 is focused on reviewing the HypergeometricDistribution.

Calculator Instructions

If you are using the TI-83/TI-84 series, there are probability functions for the binomial, Poisson, and ge-ometric. Each has a pdf and a cdf (for example binompdf and binomcdf).These functions are located in2nd DISTR. If you use, say, , you will get the table of probabilities for 0, 1, 2, ...,n. If youuse , you will get the probability ofx. If you use , you will get thecumulative probability(P (X= 0)+P (X= 1)+P (X= 2)+ ... +P (X= n)).

Assign HomeworkAssign Homework7. Suggested homework: 1 - 17 odds, 23, 33 - 37 (Binomial and Poisson).

2"Discrete Random Variables: Practice 1: Discrete Distributions" 3"Discrete Random Variables: Practice 2: Binomial Distribution" 4"Discrete Random Variables: Practice 3: Poisson Distribution" 5"Discrete Random Variables: Practice 4: Geometric Distribution" 6"Discrete Random Variables: Practice 5: Hypergeometric Distribution" 7"Discrete Random Variables: Homework"


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14 CHAPTER 5. CH. 4: DISCRETE RANDOM VARIABLES


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Chapter 6

Ch. 5: Continuous Random Variables1

This chapter is a good introduction to continuous types of probability distributions (the most famous of allis the normal). Two continuous distributions are covered the uniform (or rectangular) and the exponential.

For the uniform, probability is just the area of a rectangle. This distribution easily gets across the conceptthat probability is equal to area under a "curve" (a function). The exponential, which is used in industryand models decay, is a nice lead-in to the normal. The uniform and exponential distributions are also nicedistributions to start with when you teach the Central Limit Theorem. It is interesting to note that theamount of money spent in one trip to the supermarket follows an exponential distribution. Several of ourstudents discovered this idea when they chose data for their second project.

Compare Binomial v. Continuous DistributionBegin this chapter by a comparison of a binomial (discrete) distribution and a continuous distribution.Using the normal for this comparison works well because the students are already familiar with it. The

binomial graph has probability = height and the normal graph has probability = area. Tell the students thatthe discovery of probability = area in the continuous graph comes from calculus (which most of them havenot studied). Draw the two graphs to make these ideas clear.

Introduce Uniform DistributionIntroduce the uniform distribution using the following example: The amount of time a student waits in lineat the college cafeteria is uniformly distributed in the interval from 0 to 5 minutes (the students must waitin line from 0 to 5 minutes - each time in this interval is equally likely). Note: all the times cannot be listed.This is different from the discrete distributions.

Example 6.1LetX= the amount of time (in minutes) a student waits in line at the college cafeteria. The notation

for the distribution is X U(a, b) where a = 0 and b = 5. The function is f( x) = 15 where0 < x < 5. The pattern is f(x)= 1ba wherea < x< b.

In this examplea =0 andb =5. The function f( x)where 0 < x < 5 graphs as a horizontal line

segment.1This content is available online at .

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16 CHAPTER 6. CH. 5: CONTINUOUS RANDOM VARIABLES

Figure 6.1: Because 0 < x < 5, the maximum area = (15)(5)=1, the largest probability possible.

Example 6.2Find the probability that a student must wait less than 3 minutes. Draw the picture and write the

probability statement.

Solution

Figure 6.2:Probability statement: P(X


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Example 6.4Find the 75th percentile of waiting times. A time is being asked for here. Percentiles often confuse

students. They see "75th" and think they need to find a probability. Draw a picture and write aprobability statement. Letk= the 75th percentile.

Solution

Figure 6.3

Probability statement: P (X< k)= 0.75 Area:(k 0)

15

= 0.75 k= 3.75 minutes

75% of the students wait at most 3.75 minutes and 25% of the students wait at least 3.75 minutes.

Example 6.5

You can finish the uniform with a conditional. This reviews conditionals from Continuous Ran-dom Variables2. What is the probability that a student waits more than 4 minutes when he/shehas already waited more than 3 minutes?

SolutionAlgebraically:P (X> 4|X> 3)= P(X>4 AND X>3)P(X>3) =

P(X>4)P(X>3)

NOTE: The students see it more clearly if you do the problem graphically. The lower value, a,changes from 0 to 3. The upper value stays the same(b= 5). The function changes to: f(x) =

153 =

12

2"Continuous Random Variables: Introduction"


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Figure 6.4: P (X> 4|X> 3)=(base) (height)=(5 4)

12

= 12

Introduce the Change ExampleThe exponential distribution is generally concerned with how a quantity declines or decays. Examplesinclude the life of a car battery, the life of a light bulb, the length of time business long distance telephonecalls last, and the amount of change a person is carrying. You can introduce the exponential by using thechange example. Ask everyone in your classroom to count their change and record it. Then have themcalculate the mean and standard deviation and graph the histogram. The histogram should appear to

be declining. Let X= the amount of change one person carries. Notation: X Exp (m) where m is theparameter that controls the amount of decline or decay; m = 1 and =

1m . Also, = . (In the example,

the calculated mean and standard deviation ought to be fairly close.)

Example 6.6The function is where f( x)= memxm 0 ANDx 0. Find the probability that the amount of

change one person has is less then $.50. Draw the graph.

Figure 6.5:The right tail extends indefinitely. There is no upper limit in x.

The formula is P (X< x) = 1 emx P (X< .50) =_________. The authors use technology tosolve the probability problems. If you use the TI-83/84 calculator series, enter on the home-screen,

1 em.50. Fill in them with whatever the data produces ( m = 1 ; replace with the samplemean).


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19

Ask the question, "Ninety percent of you have less than what amount?" and have them find the90th percentile.

Draw the picture and letk= the 90th percentile. P (X< k)= 0.90. Solve the equation 1

emk =

0.90 fork. On the home-screen of the TI-83/TI-84, enter ln(1.90)(m) .

NOTE: Have students fill in the blanks.

On average, a student wouldexpectto have _________ . The word "expect" implies the mean. Tenstudents together wouldexpectto have _________. (the mean multiplied by 10)

Assign PracticeAssign the Practice 13 and Practice 24 in class to be done in groups.

Assign HomeworkAssign Homework5 . Suggested problems: 1 -13 odds, 15 - 20.

3"Continuous Random Variables: Practice 1" 4"Continuous Random Variables: Practice 2" 5"Continuous Random Variables: Homework"


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27/59

Chapter 7

CH. 6: Normal Distribution1

A fair number of students are familiar with the "bell-shaped" curve. Stress that the normal is a continuousdistribution like the uniform and exponential. However, the left and right tails extend indefinitely but come

infinitely close to thex-axis. It is not necessary to show the probability distribution function for the normal(it is in the book) because there are normal probability tables and technology available for probability andpercentile calculations.

Visualize the DataDraw a picture of the normal graph and explain that it is symmetrical about the mean. The shape of thegraph depends on the standard deviation. The smaller the standard deviation, the skinnier and taller thegraph. A change in the mean shifts the graph to the right or left. The notation for the normal is X~N(, ).Draw several normal curves (superimposed upon each other). Have students determine how the meansand standard deviations are changing.

Figure 7.1

1

This content is available online at .

21


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22 CHAPTER 7. CH. 6: NORMAL DISTRIBUTION

Figure 7.2

The Normal Distribution NotationThe standard normal distribution is of special interest. Notation: Z~N(0, 1)where Z = one z-score (thenumber of standard deviations a value is to the right or left of the mean). The mean is 0 and the variance(and standard deviation) is 1. Any normal distribution can be standardized to the standard normal by the

z-score formula: z = value . Do an example showing the standardization. ForX~N(3, 2)andY~N(5, 6),the values x = 4 and y = 8 are each 12 standard deviation to the right (

12) of their respective means.

Therefore, they both have a z-score of 12 .

Example 7.1Do an example using the normal distribution and the standardization.

ProblemSeveral studies have shown that the amount of time people stand in line waiting for a bank telleris normally distributed. Suppose the mean waiting time is 3 minutes and the standard deviation is1.5 minutes. LetX= the amount of time, in minutes, one person stands in line waiting for a teller.Notation: X~N(3,1.5)

Find the probability that one person waits in line for a teller less than 2 minutes. Have studentsdraw the picture and write a probability statement. The picture should have the x-axis.

Solution

Figure 7.3: Probability statement: P (X p-value or < p-value.

The example in the book concerning Jeffrey, an eight-year old swimmer, is a good first example to do withthe class. They can follow along in the book and then complete the problem that follows (bench pressproblem). By filling in the blanks, they are led through the steps of hypothesis testing.

In the beginning, the students have the most difficulty in determining which test to use (test of a singlemean - normal or Student-t or test a binomial proportion) and the type (left-, right-, or two-tailed). We doseveral examples (usually we choose some homework problems) in class with the students. If a single meanStudent-t is done, the assumption is that the population from which the data is taken is normal. In reality,this would have to be shown to be true.

Here

2

is a series of solution sheets that can be copied and used by the students to do the hypothesis testingproblems. A solution sheet makes it clearer to the student what the steps to the tests are.

Go over the solution for "Fidos Fleas", a binomial proportion hypothesis testing problem written as a poem.The problem is at the end of the text portion of the chapter. The solution on a solution sheet follows thepoem.

If you use the TI-83/84 series, there are functions to perform the different hypotheses tests. They can befound in STAT TESTS. Z-Test (normal test) does a test of a single mean when the population standard devi-ation is known; T-test (Student-t test) does a test of a single mean when the population standard deviationis not known; 1-PropZTest (normal test) does a test of a single proportion. The examples in the book containTI-83/84 calculator instructions, in detail.

Assign Practice

Assign Practice 13, Practice 24, and Practice 35 to be done collaboratively.

Assign HomeworkAssign Homework6. Suggested problems: 1 - 15 odds, 19, 21, 25, 29, 31, 33, 34 - 44.

Assign ProjectsThere are two partner projects for this lesson: one uses an article7 and the other is a word problem8. Stu-dents create their own hypothesis testing problems and learn much from the process.

2"Collaborative Statistics: Solution Sheets: The Chi-Square Distribution" 3"Hypothesis Testing of Single Mean and Single Proportion: Practice 1" 4"Hypothesis Testing of Single Mean and Single Proportion: Practice 2" 5"Hypothesis Testing of Single Mean and Single Proportion: Practice 3" 6"Hypothesis Testing of Single Mean and Single Proportion: Homework" 7"Collaborative Statistics: Projects: Hypothesis Testing Article" 8"Collaborative Statistics: Projects: Hypothesis Testing Word Problem"


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Chapter 11

Ch 10: Hypothesis Testing of Two Meansand Two Proportions1

The comparison of two groups is done constantly in business, medicine, and education, to name just a fewareas. You can start this chapter by asking students if they have read anything on the Internet or seen ontelevision any studies that involve two groups. Examples include diet versus hypnotism, Bufferin withaspirin versus Tylenol, Pepsi Cola versus Coca Cola, and Kelloggs Raisin Bran versus Post RaisinBran. There are hundreds of examples on the Internet, in newspapers, and in magazines.

This chapter covers independent groups for two population means and two population proportions andmatched or paired samples. The module relies heavily on technology. Instructions for the TI-83/84 seriesof calculators are included for each example. If you and your class are interested, the formulas for the teststatistics are included in the text.

Doing problems 1 - 10 in the Homework2 helps the students to determine what kind of hypothesis test they

should perform.Example 11.1: Matched or Paired SamplesA course is designed to increase mathematical comprehension. In order to evaluate the effective-

ness of the course, students are given a test before and after the course. The sample data is:

Before Course 90 100 160 112 95 190 125

After Course 120 95 150 150 100 200 120

Table 11.1

Example 11.2: Two Proportions, Independent Groups

Suppose in the last local election, among 240 30-45 year olds, 45% voted and among 260 46-60 yearolds, 50% voted. Does the data indicate that the proportion of 30-45 year olds who voted is lessthan the proportion of 46-60 year olds? Test at a 1% level of significance.

Firm A:

NA=20 SA=$100XA=$1500

1This content is available online at .2"Hypothesis Testing of Two Means and Two Proportions: Homework"

33


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34 CHAPTER 11. CH 10: HYPOTHESIS TESTING OF TWO MEANS AND TWOPROPORTIONS

Firm B:

NB=22

SB=$200XB = $1900

Test the claim that the average price of Firm As laptop is no different from the average price ofFirm Bs laptop.

Calculator InstructionsIf you use the TI83/84 series, the functions are located in STATS TESTS. The function for two proportionsis 2-PropZTest, the function for two means is 2-SampTTest if the population standard deviations are notknown and 2-SampZTest if the population standard deviations are known (highly unlikely). The functionfor matched pairs is T-test (the same test used for test of a single mean) because you combine two measure-ments for each object into a single set of "difference" data. For the function 2-SampTTest, answer "NO" to"Pooled."

Assign PracticeHave students do the Practice 13 and Practice 24 collaboratively in class. These practices are for two pro-

portions and two means. For matched pairs, you could have them do Example 10-7 in the text.

Assign HomeworkAssign Homework5. Suggested homework problems: 1 - 10, 11, 13, 15, 17, 19, 23, 25, 31, 39 - 52.

3"Hypothesis Testing: Two Population Means and Two Population Proportions: Practice 1"

4"Hypothesis Testing: Two Population Means and Two Population Proportions: Practice 2"

5"Hypothesis Testing of Two Means and Two Proportions: Homework"


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Chapter 12

Ch 11: The Chi-Square Distribution1

This chapter is concerned with three chi-square applications: goodness-of-fit; independence; and singlevariance. We rely on technology to do the calculations, especially for goodness-of-fit and for independence.However, the first example in the chapter (the number of absences in the days of the week) has the studentcalculate the chi-square statistic in steps. The same could be done for the chi-square statistic in a test ofindependence.

The chi-square distribution generally is skewed to the right. There is a different chi-square curve for eachdf. When the dfs are 90 or more, the chi-square distribution is a very good approximation to the normal.For the chi-square distribution,= the number of dfs and= the square root of twice the number of dfs.

Goodness-of-Fit TestA goodness-of-fit hypothesis test is used to determine whether or not data "fit" a particular distribution.

Example 12.1In a past issue of the magazine GEICO Direct, there was an article concerning the percentage

of teenage motor vehicle deaths and time of day. The following percentages were given from asample.

Time of Day Percentage of Motor Vehicle Deaths

Time of Day Death Rate

12 a.m. to 3 a.m. 17%

3 a.m. to 6 a.m. 8%

6 a.m. to 9 a.m. 8%

9 a.m. to 12 noon 6%

12 noon to 3 p.m. 10%

3 p.m. to 6 p.m. 16%

6 p.m. to 9 p.m. 15%

9 p.m. to 12 a.m. 19%

Table 12.1


35


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36 CHAPTER 12. CH 11: THE CHI-SQUARE DISTRIBUTION

For the purpose of this example, suppose another sample of 100 produced the same percentages.We hypothesize that the data from this new sample fits a uniform distribution. The level of signif-icance is 1% (= 0.01 ).

Ho: The number of teenage motor vehicle deaths fits a uniform distribution. Ha: The number of teenage motor vehicle deaths does not fit a uniform distribution.

The distribution for the hypothesis test is X27

The table contains the observed percentages. For the sample of 100, the observed (O) numbers are17, 8, 8, 6, 10, 16, 15 and 19. The expected (E) numbers are each 12.5 for a uniform distribution (100divided by 8 cells). The chi-square test statistic is calculated using

8(0E)2

E

= (1712.5)2

12.5 + (812.5)2

12.5 + (812.5)2

12.5 + (612.5)2

12.5 + (1012.5)2

12.5 + (1612.5)2

12.5 + (1512.5)2

12.5 + (1912.5)2

12.5

=13.6

If you are using the TI-84 series graphing calculators, ON SOME OF THEM there is a function inSTAT TESTS calledx2 GOF-Test that does the goodness-of-fit test. You first have to enter the ob-served numbers in one list (enter as whole numbers) and the expected numbers (uniform impliesthey are each 12.5) in a second list (enter 12.5 for each entry: 100 divided by 8 = 12.5). Then do thetest by going tox2 GOF-Test.

If you are using the TI-83 series, enter the observed numbers in list1 and the expected numbers inlist2 and in list3 (go to the list name), enter (list1-list2)^2/list2. Press enter. Add the values in list3(this is the test statistic). Then go to 2nd DISTRx2cdf. Enter the test statistic (13.6) and the uppervalue of the area (10^99) and the degrees of freedom (7).

Probability Statement: P x2 > 13.6= 0.0588(Always a right-tailed test)

Figure 12.1: p-value= 0.0588

Since < p-value(0.01 < 0.0588), we do not rejectHo.

We conclude that there is not sufficient evidence to reject the null hypothesis. It appears that thenumber of teenage motor vehicle deaths fits a uniform distribution. It does not matter what timeof the day or night it is. Teenagers die from motor vehicle accidents equally at any time of the day


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37

or night. However, if the level of significance were 10%, we would reject the null hypothesis andconclude that the distribution of deaths does not fit a uniform distribution.

Atest of independencecompares two factors to determine if they are independent (i.e. one factor does not

affect the happening of a second factor).Example 12.2

The following table shows a random sample of 100 hikers and the area of hiking pre-ferred.

Hiking Preference Area

Gender The Coastline Near Lakes and Streams On Mountain Peaks

Female 18 16 11

Male 16 25 14

Table 12.2: The two factors are gender and preferred hiking area.

Ho: Gender and preferred hiking area are independent. Ha: Gender and preferred hiking area are not independent

The distribution for the hypothesis test is x22.

The dfs are equal to: (rows 1) (columns 1) = (2 1) (3 1) = 2

The chi-square statistic is calculated using (23)(0E)2

E

Each expected (E) value is calculated using (rowtotal)(columntotal)

totalsurveyed

The first expected value (female, the coastline) is 4534100 =15.3The expected values are: 15.3, 18.45, 11.25, 18.7, 22.55, 13.75

The chi-square statistic is:

(23)(0E)2

E =

(1815.3)215.3 +

(1618.45)218.45 +

(1111.15)211.25 +

(1618.7)218.7 +

(2522.55)222.55 +

(1413.75)213.75

=1.47

Calculator Instructions

The TI-83/84 series have the functionx

2

-Test in STAT TESTS to preform this test. First, you haveto enter the observed values in the table into a matrix by using 2nd MATRIX and EDIT [A]. Enterthe values and go to x2-Test. Matrix [B] is calculated automatically when you run the test.

Probability Statement: p-value= 0.4800 (A right-tailed test)


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Sinceis less than 0.05, we do not reject the null.

There is not sufficient evidence to conclude that gender and hiking preference are not independent.Sometimes you might be interested in how something varies. A test of a single variance is the type ofhypothesis test you could run in order to determine variability.

Example 12.3A vending machine company which produces coffee vending machines claims that its machine

pours an 8 ounce cup of coffee, on the average, with a standard deviation of 0.3 ounces. A collegethat uses the vending machines claims that the standard deviation is more than 0.3 ounces causingthe coffee to spill out of a cup. The college sampled 30 cups of coffee and found that the standarddeviation was 1 ounce. At the 1% level of significance, test the claim made by the vending machinecompany.

SolutionHo :

2 =(0.3)2 Ha : 2 > (0.3)2

The distribution for the hypothesis test is x229where df= 30 1= 29.

The test statisticx2 = (n1)s2

2 = (301)1

2

0.32 =322.22

Probability Statement: P

x2 > 322.22

= 0

Figure 12.3: p-value= 0

Sincea > p-value(0.01 > 0), rejectHo.

There is sufficient evidence to conclude that the standard deviation is more than 0.3 ounces ofcoffee. The vending machine company needs to adjust their machines to prevent spillage.


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39

Assign Practice

Have the students do the Practice 12

, Practice 23

, and Practice 34

in class collaboratively.

Assign HomeworkAssign Homework5 . Suggested homework: 3, 5, 7 (GOF), 9, 13, 15 (Test of Indep.), 17, 19, 23 (Variance), 24- 37 (General)

2"The Chi-Square Distribution: Practice 1" 3"The Chi-Square Distribution: Practice 2" 4"The Chi-Square Distribution: Practice 3" 5"The Chi-Square Distribution: Homework"


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47/59

Chapter 13

Ch 12: Linear Regression andCorrelation1

Entire courses are given on linear regression and correlation. This chapter serves as an introduction to thetopics.

It helps to review the equation of a line. We use a for they-intercept andb for the slope. The line has theform:y = a +bx

Example 13.1Have the students plot a line by eye using the following data. The independent variable x repre-

sents the size of a color television screen in inches at Andersons and y represents the sales price

in dollars.

x 9 20 27 31 35 40 60

y 147 197 297 447 1177 2177 2497

Table 13.1

Ask them what they got for the slope and for the y-intercept. Make comparisons. This exerciseshould point out how difficult it is to get an accurate line of best fit and how many lines "seem" tofit the data. (This data is taken from the exercises.)

SolutionFor the data above, use either a calculator or a computer and calculate the least squares or best fitline. Look at the scatter plot first. Ask the students if their "by eye" line looks like the calculatedone. Explain the correlation coefficient and then check if the correlation coefficient is significant bycomparing it to the correct entry in 95% CRITICAL VALUES OF THE SAMPLE CORRELATION

COEFFICIENT Table at the end of the reading.

If you use the TI-83/84 series, enter the data into two lists first. Then plot the data points on thecalculator. First set up the stat plot (2nd STAT PLOT). Then press ZOOM 9 to see the plot. To dothe linear regression, go to the LinReg ( a +bx) function in STAT CALC. Enter the lists. At thistime, you could also enter a y-variable after the lists (after you enter the lists, enter a comma andthen press VARS Y-VARS Function Y1). Press ENTER to see the linear regression. When you pressGRAPH, the line will plot.


41


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42 CHAPTER 13. CH 12: LINEAR REGRESSION AND CORRELATION

Line of best fit: yhat= 745.2420 + 54.7557x.

Explain "predicting" (or forecasting) and have them predict the sales price of a 45 inch screen color TV. Havethem predict the cost for a mini 5 inch color TV. (The answer is negative.) Discuss that the line is only validfrom the lowest to the highest x - values.

Example 13.2Have the students follow the "outlier" example in the text and (just once!) do the calculations forfinding an outlier. Have them fill in the table below.

x y y yhat | y yhat | (| y yhat | )2

Table 13.2

Find: 7(| y yhat | )2 =SSE

Finds =

SSEn2

n=the total number of data values (7 for this problem)

sis the standard deviation of the | y yhat | valuesMultiplysby 1.9:(1.9) (s)=_______

Compare each | y yhat | to(1.9) (s).If any

| y

yhat

|is at least (1.9) (s), then the corresponding point is an outlier. (None of the

points is an outlier.)

Assign PracticeHave the students do the Practice2 collaboratively in class.

Assign HomeworkAssign Homework.3 Suggested homework: 1, 3, 5, 9, 13, 15 (a - f only if you use the calculator), 21 - 25.

2"Linear Regression and Correlation: Practice" 3"Linear Regression and Correlation: Homework"


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Chapter 14

Ch 13: F Distribution and ANOVA1

HISTORY: The F distribution is named after Ronald Fisher. Fisher is one of the most respectedstatisticians of all time. He did a lot of statistical work in biology and genetics and became chair

of genetics at Cambridge University in England in 1949. In 1952, he was awarded knighthood.

This section is avery brief overviewof theFdistribution and two of its applications - One Way Analysis ofVariance (ANOVA) and test of two variances. There are college courses which deal exclusively with thesetopics. ANOVA, particularly, is used regularly in industry.

Explanation of Sum of Squares, Mean Square, and the F ratio for ANOVA

k= the number of different groups nj= the size of the jth group sj= the sum of the values in the jth group N= the total number of all the values combined Total sample size: nj

x= one value: x= sj

Sum of squares of all values from every group combined:x2 Between group variability: SStotal = x2 (x)

2

N

Total sum of squares:x2 (x)2

N Explained variation- sum of squares representing variation among the different samples SSbetween =

(sj)2

nj

(sj)

2

N

Unexplained variation- sum of squares representing variation within samples due to chance:SSwithin= SStotal SSbetween

dfs for different groups (dfs for the numerator): dfbetween=k 1 Equation for errors within samples (dfs for the denominator): dfwithin= N k Mean square (variance estimate) explained by the different groups: MSbetween= SSbetweendfbetween Mean square (variance estimate) that is due to chance (unexplained): MSwithin= SSwithindfwithin F ratio or F statistic of two estimates of variance:F = MSbetweenMSwithin

NOTE: The above calculations were done with groups of different sizes. If the groups are the samesize, the calculations simplify somewhat and the F ratio can be written as:


43


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44 CHAPTER 14. CH 13: F DISTRIBUTION AND ANOVA

F Ratio Formula

F= n (sx)2

spooled2

(14.1)

where...

(sx)2 =the variance of the sample means n=the sample size of each group

spooled

2=the mean of the sample variances (pooled variance)

dfnumerator= k 1 dfdenominator= k(n 1)= N k

These calculations are easily done with a graphing calculator or a computer program. We present theinformation in the chapter assuming some kind of technology will be used.

For ANOVA, the samples must come from normally distributed populations with the same variance, and

the samples must be independent. The ANOVA test is right-tailed.

In a test of two variances, the samples must come from normal populations and must be independent ofeach other.

Exercise 14.1 (Solution on p. 46.)(One-Way ANOVA)Three different diet plans are to be tested for average weight loss. For each diet plan, 4 dieters are

selected and their weight loss (in pounds) in one months time is recorded.

Plan 1 Plan 2 Plan 3

5 3.5 8

4.5 7 4

4 6 3.5

3 4 4.5

Table 14.1

Is the average weight loss the same for each plan? Conduct an ANOVA test with a 1% level ofsignificance.

Exercise 14.2 (Solution on p. 47.)(Test of Two Variances):Machine A makes a box and machine B makes a lid. For the lid to fit the box correctly, the

variances should be nearly the same. There is a suspicion that the variance of the box is greater

than the variance of the lid. The following data was collected.

Machine A (Box) Machine B (Lid)

Number of Parts 9 11

Variance 150 45

Table 14.2


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45

Are the machines working properly? Test at a 5% level of significance.

Assign PracticeHave the students work collaboratively to complete the Practice2.

Assign HomeworkAssign Homework3. Suggested homework: 1, 3, 4, 5.

2"F Distribution and ANOVA: Practice" 3"F Distribution and ANOVA: Homework"


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Solutions to Exercises in Chapter 14

Solution to Exercise 14.1 (p. 44)Let1,2, and3be the population means for the three diet plans.

Ho : 1= 2 = 3 Ha : Not all pairs of means are equal.

dfnumerator= 3 1= 2 dfdenominator= 12 3= 9

The distribution for the test is F2,9

Using a calculator or computer, the test statistic is F = 0.47. The notation used for the F statistic may alsobeF or F2,9(like the distribution). The TI-83/84 series has the function ANOVA in STAT TESTS. Enter thelists of data separated by commas.

If you use the formulas for groups of the same size, the calculations are as follows:

Sample means are 4.13, 5.13, and 5, respectively. Sample standard deviations are 0.8539, 1.6250, and 2.0412,respectively.

(sx)2 =0.2956 The variance of the sample meansspooled

2= 2.5416 The mean of the sample variances

n= 4 The sample size of each group

Table 14.3

F=4 0.2956

2.5416 (14.2)

Probability Statement: P (F > 0.47)= 0.6395


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47


Since B2

nA=9 nB=11

dfnumerator= 9 1 dfdenominator= 11 1= 10

The distribution for the hypothesis test is F8,10

If you are using the TI-83/84 calculators, use the function 2-SAMPFTest for the test.

Using the formulas,


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The test statistic is F =

(sA)

2

(A)

2

(sB)

2

(B)

2 =

(sA)2

(sB)

2

= 15045 =3.33

Since > p-value, reject the null hypothesis.

There is sufficient evidence to conclude that the box and lid do not fit each other. The variance of the box islarger.


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INDEX 49

Index of Keywords and Terms

Keywordsare listed by the section with that keyword (page numbers are in parentheses). Keywordsdo not necessarily appear in the text of the page. They are merely associated with that section. Ex.apples, 1.1 (1)Termsare referenced by the page they appear on. Ex. apples, 1

B binomial, 5(11)

C Collaborative, 1(1)continuous, 6(15)Course, 1(1)

D descriptive, 3(5)discrete, 5(11)distribution, 5(11), 6(15)

E Elementary, 1(1), 2(3), 3(5), 4(7), 5(11), 6(15), 7(21), 8(25), 9(27), 10(31), 11(33), 12(35), 13(41), 14(43)exponential, 6(15)

F function, 5(11), 6(15)

G geometric, 5(11)

Guide, 1(1), 3(5), 4(7), 5(11), 6(15)

H hypergeometric, 5(11)

P Plan, 1(1)Poisson, 5(11)probability, 4(7), 5(11), 6(15)

R random, 5(11), 6(15)

S Statistics, 1(1), 2(3), 3(5), 4(7), 5(11), 6(15), 7(21), 8(25), 9(27), 10(31), 11(33), 12(35), 13(41), 14(43)

T Teacher, 1(1), 3(5), 4(7), 5(11), 6(15)

U uniform, 6(15)

V variable, 5(11), 6(15)


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50 ATTRIBUTIONS

Attributions

Collection: Collaborative Statistics Teachers Guide

Edited by: Barbara Illowsky, Ph.D., Susan DeanURL: http://cnx.org/content/col10547/1.5/License: http://creativecommons.org/licenses/by/2.0/

Module: "Collaborative Statistics: Teachers Guide: Suggested Plan for Teaching the Course"Used here as: "Suggested Plan for Teaching the Course"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m17214/1.6/Pages: 1-2Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/

Module: "Sampling and Data: Teachers Guide"Used here as: "Ch. 1: Sampling and Data"

By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m16130/1.10/Pages: 3-4Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/

Module: "Descriptive Statistics: Teachers Guide"Used here as: "Ch. 2: Descriptive Statistics"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m16802/1.10/Pages: 5-6Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/

Module: "Probability Topics: Teachers Guide"Used here as: "Ch. 3: Probability Topics"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m16844/1.11/Pages: 7-9Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/

Module: "Discrete Random Variables: Teachers Guide"Used here as: "Ch. 4: Discrete Random Variables"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m16834/1.13/

Pages: 11-13Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/


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ATTRIBUTIONS 51

Module: "Continuous Random Variables: Teachers Guide"Used here as: "Ch. 5: Continuous Random Variables"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m16814/1.12/


Module: "Normal Distribution: Teachers Guide"Used here as: "CH. 6: Normal Distribution"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m16990/1.9/Pages: 21-23Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/

Module: "Central Limit Theorem: Teachers Guide"

Used here as: "Ch. 7: Central Limit Theroem"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m16957/1.8/Pages: 25-26Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/

Module: "Confidence Intervals: Teachers Guide"Used here as: "Ch. 8: Confidence Intervals"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m16975/1.11/Pages: 27-30Copyright: Maxfield Foundation

License: http://creativecommons.org/licenses/by/2.0/Module: "Hypothesis Testing: Single Mean and Single Proportion: Teachers Guide"Used here as: "Ch. 9: Hypothesis Testing of Single Mean and Single Proportion"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m17008/1.8/Pages: 31-32Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/

Module: "Hypothesis Testing: Two Population Means and Two Population Proportions: Teachers Guide"Used here as: "Ch 10: Hypothesis Testing of Two Means and Two Proportions"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m17020/1.8/



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52 ATTRIBUTIONS

Module: "The Chi-Square Distribution: Teachers Guide"Used here as: "Ch 11: The Chi-Square Distribution"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m17060/1.11/


Module: "Linear Regression and Correlation: Teachers Guide"Used here as: "Ch 12: Linear Regression and Correlation"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m17084/1.10/Pages: 41-42Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/

Module: "F Distribution and ANOVA: Teachers Guide"

Used here as: "Ch 13: F Distribution and ANOVA"By: Susan Dean, Barbara Illowsky, Ph.D.URL: http://cnx.org/content/m17073/1.9/Pages: 43-48Copyright: Maxfield FoundationLicense: http://creativecommons.org/licenses/by/2.0/


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Collaborative Statistics Teachers GuideThis collection is the companion teachers guide for Collaborative Statistics (col10522) by Barbara Illowskyand Susan Dean.

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Collaborative Statistics Teacher's Guide

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